Ylm.ipynb

{
 "metadata": {
  "name": "Ylm"
 },
 "name": "Ylm",
 "nbformat": 3,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "%load_ext sympyprt",
     "language": "python",
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from sympy import *",
     "language": "python",
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "theta = Symbol(\"theta\")\nphi = Symbol(\"phi\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(0, 5):\n    for j in range(-i, i+1):\n        print((i, j, simplify(Ylm(i, j, theta, phi))))",
     "language": "python",
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "(0, 0, legendre(0, cos(theta))/(2*sqrt(pi)))\n(1, -1, sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi)))\n(1, 0, sqrt(3)*legendre(1, cos(theta))/(2*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(1, 1, -sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi)))\n(2, -2, sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, -1, sqrt(30)*exp(-I*phi)*sin(theta)*cos(theta)/(4*sqrt(pi)))\n(2, 0, sqrt(5)*legendre(2, cos(theta))/(2*sqrt(pi)))\n(2, 1, -sqrt(30)*exp(I*phi)*sin(theta)*cos(theta)/(4*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, 2, sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi)))\n(3, -3, sqrt(35)*exp(-3*I*phi)*sin(theta)**3/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, -2, sqrt(210)*exp(-2*I*phi)*sin(theta)**2*cos(theta)/(8*sqrt(pi)))\n(3, -1, sqrt(21)*(-5*sin(theta)**2 + 4)*exp(-I*phi)*sin(theta)/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 0, sqrt(7)*legendre(3, cos(theta))/(2*sqrt(pi)))\n(3, 1, sqrt(21)*(5*sin(theta)**2 - 4)*exp(I*phi)*sin(theta)/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 2, sqrt(210)*exp(2*I*phi)*sin(theta)**2*cos(theta)/(8*sqrt(pi)))\n(3, 3, -sqrt(35)*exp(3*I*phi)*sin(theta)**3/(8*sqrt(pi)))\n(4, -4, 3*sqrt(70)*exp(-4*I*phi)*sin(theta)**4/(32*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -3, 3*sqrt(35)*exp(-3*I*phi)*sin(theta)**3*cos(theta)/(8*sqrt(pi)))\n(4, -2, 3*sqrt(10)*(-7*sin(theta)**2 + 6)*exp(-2*I*phi)*sin(theta)**2/(16*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -1, 3*sqrt(5)*(-7*sin(theta)**2 + 4)*exp(-I*phi)*sin(theta)*cos(theta)/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 0, 3*legendre(4, cos(theta))/(2*sqrt(pi)))\n(4, 1, 3*sqrt(5)*(7*sin(theta)**2 - 4)*exp(I*phi)*sin(theta)*cos(theta)/(8*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 2, 3*sqrt(10)*(-7*sin(theta)**2 + 6)*exp(2*I*phi)*sin(theta)**2/(16*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 3, -3*sqrt(35)*exp(3*I*phi)*sin(theta)**3*cos(theta)/(8*sqrt(pi)))\n(4, 4, 3*sqrt(70)*exp(4*I*phi)*sin(theta)**4/(32*sqrt(pi)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": ""
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "l = Symbol(\"l\", integer=True)\nm = Symbol(\"m\", integer=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(l, m, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
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       "prompt_number": 6,
       "text": "\n    ____________________                                    \n   \u2571 (2\u22c5l + 1)\u22c5(l - m)!   \u2148\u22c5m\u22c5\u03c6                             \n  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f     \u22c5assoc_legendre(l, m, cos(\u03b8))\n\u2572\u2571        (l + m)!                                          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                              ___                           \n                          2\u22c5\u2572\u2571 \u03c0                            "
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAW8AAAA5BAMAAAALsrd5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nET+sxUVpROdENQLEJZT/Sv3v7N8dOPeOdqtalAAAAABJRU5ErkJggg==\n",
       "prompt_number": 7,
       "text": "\n    ____________________                                    \n   \u2571 (2\u22c5l + 1)\u22c5(l - m)!   \u2148\u22c5m\u22c5\u03c6                             \n  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f     \u22c5assoc_legendre(l, m, cos(\u03b8))\n\u2572\u2571        (l + m)!                                          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                              ___                           \n                          2\u22c5\u2572\u2571 \u03c0                            "
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(-l, m, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZMAAAA5BAMAAAABwttuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nV0gnkl9t98QAAAAASUVORK5CYII=\n",
       "prompt_number": 8,
       "text": "\n    ______________________                                     \n   \u2571 (-2\u22c5l + 1)\u22c5(-l - m)!   \u2148\u22c5m\u22c5\u03c6                              \n  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f     \u22c5assoc_legendre(-l, m, cos(\u03b8))\n\u2572\u2571        (-l + m)!                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                ___                            \n                            2\u22c5\u2572\u2571 \u03c0                             "
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_, deep=True)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZMAAAA5BAMAAAABwttuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nV0gnkl9t98QAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": "\n    ______________________                                     \n   \u2571 (-2\u22c5l + 1)\u22c5(-l - m)!   \u2148\u22c5m\u22c5\u03c6                              \n  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f     \u22c5assoc_legendre(-l, m, cos(\u03b8))\n\u2572\u2571        (-l + m)!                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                ___                            \n                            2\u22c5\u2572\u2571 \u03c0                             "
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(l, -m, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfUAAAA6BAMAAABbr933AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nA0SecBeV/Af5cK6/38EBcQAAAABJRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": "\n          ____________________                                              \n    m    \u2571 (2\u22c5l + 1)\u22c5(l + m)!   -\u2148\u22c5m\u22c5\u03c6                                      \n(-1) \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f      \u22c5assoc_legendre(l, m, cos(\u03b8))\u22c5(l - m)!\n      \u2572\u2571        (l - m)!                                                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  ___                                       \n                              2\u22c5\u2572\u2571 \u03c0 \u22c5(l + m)!                              "
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_, deep=True)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfUAAAA6BAMAAABbr933AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nA0SecBeV/Af5cK6/38EBcQAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": "\n          ____________________                                              \n    m    \u2571 (2\u22c5l + 1)\u22c5(l + m)!   -\u2148\u22c5m\u22c5\u03c6                                      \n(-1) \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f      \u22c5assoc_legendre(l, m, cos(\u03b8))\u22c5(l - m)!\n      \u2572\u2571        (l - m)!                                                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  ___                                       \n                              2\u22c5\u2572\u2571 \u03c0 \u22c5(l + m)!                              "
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(-l, -m, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAicAAAA6BAMAAACZqeSCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 12,
       "text": "\n          ______________________                                              \n    m    \u2571 (-2\u22c5l + 1)\u22c5(-l + m)!   -\u2148\u22c5m\u22c5\u03c6                                      \n(-1) \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f      \u22c5assoc_legendre(-l, m, cos(\u03b8))\u22c5(-l - m\n      \u2572\u2571        (-l - m)!                                                     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                   ___                                        \n                               2\u22c5\u2572\u2571 \u03c0 \u22c5(-l + m)!                              \n\n  \n  \n)!\n  \n\u2500\u2500\n  \n  "
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_, deep=True)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAicAAAA6BAMAAACZqeSCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 13,
       "text": "\n          ______________________                                              \n    m    \u2571 (-2\u22c5l + 1)\u22c5(-l + m)!   -\u2148\u22c5m\u22c5\u03c6                                      \n(-1) \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u212f      \u22c5assoc_legendre(-l, m, cos(\u03b8))\u22c5(-l - m\n      \u2572\u2571        (-l - m)!                                                     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                   ___                                        \n                               2\u22c5\u2572\u2571 \u03c0 \u22c5(-l + m)!                              \n\n  \n  \n)!\n  \n\u2500\u2500\n  \n  "
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "Y = Ylm(l, m, theta, phi)",
     "language": "python",
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from random import random",
     "language": "python",
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rat = pi*random()\nrap = 2*pi*random()\n(rat, rap)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 16,
       "text": "(0.582321280766733\u22c5\u03c0, 0.258332267787315\u22c5\u03c0)"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(0, 5):\n    for j in range(-i, i+1):\n        y = Y.subs(l, i).subs(m, j)\n        Z = simplify(trigsimp(y - Ylm(i, j, theta, phi)))\n        print((i, j, Z.subs(theta, rat).subs(phi,rap)))",
     "language": "python",
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "(0, 0, 0)\n(1, -1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(1, 0, 0)\n(1, 1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, -2, 0)\n(2, -1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, 0, 0)\n(2, 1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, 2, 0)\n(3, -3, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, -2, 0)\n(3, -1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 0, 0)\n(3, 1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 2, 0)\n(3, 3, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -4, 0)\n(4, -3, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -2, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 0, 0)\n(4, 1, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 2, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 3, 0)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 4, 0)"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(3, 1, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQYAAAAvBAMAAADzz3lFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 18,
       "text": "\n        \u239b      2       \u239e            \n   ____ \u239c15\u22c5cos (\u03b8)   3\u239f  \u2148\u22c5\u03c6       \n-\u2572\u2571 21 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u239f\u22c5\u212f   \u22c5sin(\u03b8)\n        \u239d    2        2\u23a0            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   ___              \n              12\u22c5\u2572\u2571 \u03c0               "
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOoAAAAvBAMAAAAfn0t6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndpS3aiY7ymUCmewoF1W4/0HmL1HqrPChkzjzAAAAAElFTkSuQmCC\n",
       "prompt_number": 19,
       "text": "\n  ____ \u239b     2       \u239e  \u2148\u22c5\u03c6       \n\u2572\u2571 21 \u22c5\u239d5\u22c5sin (\u03b8) - 4\u23a0\u22c5\u212f   \u22c5sin(\u03b8)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 ___              \n             8\u22c5\u2572\u2571 \u03c0               "
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(3, (1.5+I)/3.0, pi/6.0, 1.0).evalf()",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 24,
       "text": "\n                                                             0.5              \n0.469318037163628\u22c5(0.0779763116886809 - 0.0389578067739132\u22c5\u2148)   \u22c5(3.0441149281\n\n                          0.5                                                 \n2474 - 1.17504594726731\u22c5\u2148)   \u22c5assoc_legendre(3, 0.5 + 0.333333333333333\u22c5\u2148, cos\n\n                                                                              \n(0.166666666666667\u22c5\u03c0)) + 0.256389612231091\u22c5\u2148\u22c5(0.0779763116886809 - 0.038957806\n\n          0.5                                        0.5                      \n7739132\u22c5\u2148)   \u22c5(3.04411492812474 - 1.17504594726731\u22c5\u2148)   \u22c5assoc_legendre(3, 0.5\n\n                                                 \n + 0.333333333333333\u22c5\u2148, cos(0.166666666666667\u22c5\u03c0))"
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Ylm(S(3)/2, S(1)/2, theta, phi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAzCAMAAAB2Upe4AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\niSxmAAAAAElFTkSuQmCC\n",
       "prompt_number": 25,
       "text": "\n       \u2148\u22c5\u03c6                                 \n       \u2500\u2500\u2500                                 \n  ___   2                                  \n\u2572\u2571 2 \u22c5\u212f   \u22c5assoc_legendre(3/2, 1/2, cos(\u03b8))\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      ___                  \n                  2\u22c5\u2572\u2571 \u03c0                   "
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_, deep=True)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAzCAMAAAB2Upe4AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\niSxmAAAAAElFTkSuQmCC\n",
       "prompt_number": 26,
       "text": "\n       \u2148\u22c5\u03c6                                 \n       \u2500\u2500\u2500                                 \n  ___   2                                  \n\u2572\u2571 2 \u22c5\u212f   \u22c5assoc_legendre(3/2, 1/2, cos(\u03b8))\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      ___                  \n                  2\u22c5\u2572\u2571 \u03c0                   "
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAzCAMAAAB2Upe4AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\niSxmAAAAAElFTkSuQmCC\n",
       "prompt_number": 27,
       "text": "\n       \u2148\u22c5\u03c6                                 \n       \u2500\u2500\u2500                                 \n  ___   2                                  \n\u2572\u2571 2 \u22c5\u212f   \u22c5assoc_legendre(3/2, 1/2, cos(\u03b8))\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      ___                  \n                  2\u22c5\u2572\u2571 \u03c0                   "
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "",
     "language": "python",
     "outputs": []
    }
   ]
  }
 ]
}